$\begingroup$When it comes to the error term, the moments are the only things you can reason meaningfully with. In the context of regression analysis, once you're here: $$\implies\frac{1}{ y}\frac{\partial{y}}{\partial{x}} = \beta \frac{1}{x} + \frac{\partial{\varepsilon}}{\partial{x}}$$, take the expected value of both sides and rely on your assumptions about the distribution of $\epsilon$.$\endgroup$
– hehOct 15 '19 at 14:49

$\begingroup$But those "difficulties" are precisely why the question is of any practical interest, so I'm not sure dropping the error term is the right approach. Your model should have $\hat{\epsilon}$ in it. And I think there's an abuse of notation here, as you moved from $\hat{y}$ to $y$ without taking an expected value - which is the key operation, as given a well-behaved error term, that is what makes the offending derivative vanish.$\endgroup$
– hehOct 15 '19 at 14:47

$\begingroup$@heh Agreed that my original version moved from $\hat{y}$ to $y$ without a justification (edited to correct).$\endgroup$
– Adam BaileyOct 16 '19 at 13:14

$\begingroup$Your estimator for the error really needs to stick around, though. The true model would be what you've presented without the "hats", but again - doing this analysis on the true model is just algebra. There is no need to get philosophical about differentiation of a random variable provided one uses the necessary assumptions about the error's distribution.$\endgroup$
– hehOct 17 '19 at 14:33